Problem: $f(x, y) = x^2 + y^2 - \dfrac{x^2y^2 + y^4}{x^2 + y^2}$ We have a change of variables: $\begin{aligned} x &= X_1(r, \theta) = r \cos(\theta) \\ \\ y &= X_2(r, \theta) = r \sin(\theta) \end{aligned}$ What is $f(x, y)$ under the change of variables? Choose 1 answer: Choose 1 answer: (Choice A) A $r^2 \sin^2(\theta)$ (Choice B) B $r^2 \cos^2(\theta)$ (Choice C) C $r \sin^2(\theta)$ (Choice D) D $r \cos^2(\theta)$
Solution: When applying a change of variables, we substitute the new definition for $x$ and $y$ into the original equation. The original equation: $f(x, y) = x^2 + y^2 - \dfrac{x^2y^2 + y^4}{x^2 + y^2}$ Let's substitute $X_1(r, \theta)$ for $x$ and $X_2(r, \theta)$ for $y$. We can rewrite every $x^2 + y^2$ using the trigonometric identity that $\cos^2(\theta) + \sin^2(\theta) = 1$ : $\begin{aligned} x^2 + y^2 &= r^2\cos^2(\theta) + r^2\sin^2(\theta) \\ \\ &= r^2(\cos^2(\theta) + \sin^2(\theta)) \\ \\ &= r^2 \end{aligned}$ Therefore: $\begin{aligned} f(x, y) &= x^2 + y^2 - \dfrac{(x^2 + y^2)y^2}{x^2 + y^2} \\ \\ &= r^2 - r^2\sin^2(\theta) \\ \\ &= r^2 ( 1 - \sin^2(\theta) ) \\ \\ &= r^2 \cos^2(\theta) \end{aligned}$ Therefore, under the change of variables, $f(x, y)$ becomes: $r^2 \cos^2(\theta)$